# Re: st: MFX after ZINB

 From rgutierrez@stata.com (Roberto G. Gutierrez, StataCorp.) To statalist@hsphsun2.harvard.edu Subject Re: st: MFX after ZINB Date Thu, 13 Feb 2003 16:59:39 -0600

```Chris Roebuck <Chris.Roebuck@Advancepcs.com> asks:

> Can someone please explain how one could get POSITIVE marginal effects
> (using MFX) after running a ZINB model where the corresponding coefficient
> estimate is NEGATIVE.

This can occur when the covariate exists in both equations of the model, the
main equation and the "zero inflate" equation.  Without loss of generality,
consider a model with only one covariate (x), which exists in both equations.

The main equation uses the linear predictor: e1 = b0 + b1*x.
The inflation equation uses the linear predictor:  e2 = a0 + a1*x.

In which case, the predicted value used by default in -mfx- is:

exp(e1bar)
nhat =   --------------------
1 + exp(e2bar)

where e1bar is e1 with x replaced by the mean of the x's and e2bar is similarly
defined.

The marginal effect is the derivative of nhat with respect to x, evaluated at
x = xbar (default for -mfx-).  In this case,

nhat'(xbar) = exp(e1bar)*exp(e2bar)*(b1 - a1) + exp(e1bar)*b1
-----------------------------------------------
(1 + exp(e2bar))^2

The denominator is always positive, so that doesn't affect the sign of the
marginal effect.  Note, however, the factor (b1 - a1) in the numerator.  It is
because of this factor that in certain situations you can have a1 and b1
negative, but a positive marginal effect.  This occurs when b1 is sufficiently
greater than a1 so as to "negate" the effect of the second term in the
numerator.

--Bobby
rgutierrez@stata.com
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